Journal of Modern Physics, 2013, 4, 930-939
http://dx.doi.org/10.4236/jmp.2013.47125 Published Online July 2013 (http://www.scirp.org/journal/jmp)
Mathematical Derivation of Angular Momenta in
Quantum Physics
Daniel Grucker
Université de Strasbourg-CNRS, Strasbourg, France
Received April 24, 2013; revised May 27, 2013; accepted June 26, 2013
Copyright © 2013 Daniel Grucker. This is an open access article distributed under the Creative Commons Attribution License, which
permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ABSTRACT
For a two-dimensional complex vector space, the spin matrices can be calculated directly from the angular momentum
commutator definition. The 3 Pauli matrices are retrieved and 23 other triplet solutions are found. In the three-dimen-
sional space, we show that no matrix fulfills the spin equations and preserves the norm of the vectors. By using a Clif-
ford geometric algebra it is possible in the four-dimensional spacetime (STA) to retrieve the 24 different spins 1/2. In
this framework, spins 1/2 are rotations characterized by multivectors composed of 3 vectors and 3 bivectors. Spins 1
can be defined as rotations characterized by 4 vectors, 6 bivectors and 4 trivectors which result in unit multivectors
which preserve the norm. Let us note that this simple derivation retrieves the main spin properties of particle physics.
Keywords: Quantum Systems; Spin 1/2; Spin 1; Particle Physics, Spacetime Algebra
1. Introduction 0101 0
,,
1000 1
xy z
i
i
 

 
 
 
The spin of particles was discovered by Wolfgang Pauli
toward the end of 1924 (see Fröhlich’s paper for an up-
to-date discussion of spins [1]). In quantum mechanics
textbooks, the matrix representation of spin is obtained
from the commutator general definition of the angular
momentum:
,
x
yz
J
JiJ




,
yz x
(1)
J
JiJ



 (2)


,
z
xy
J
JiJ
2222
(3)
These 3 equations are called the spin equations.
The general theory of quantum angular momenta [2]
uses the operator
x
yz
J
JJJ, the two shift op-
erators x
,
x
J
JiJ



J JiJ

and the positive ei-
genvalue

1jj of J2. The values for j are
0,12,1,32, 2,. For j = 0, the spin space is of dimen-
sion 1 and reduced to a scalar equal to zero. For j = 1/2,
the spin space is of dimension 2 and the matrix represen-
tation is given by
11
22
1
,,
2
x
xy
JJ
yz z
J

  (4)
where i

are the Pauli matrices:

 
,,
(5)
These three J matrices describe the so called spin 1/2
quantum system.
The aim of this work is to address two questions: can
the Pauli matrices or more generally any spin system be
retrieved directly from the angular momentum definition
and are the 3 Pauli matrices the only 2-dimensional quan-
tum angular momenta?
2. Properties of the Minimum-Dimension
Quantum Spin System
The smallest quantum angular momentum is of dimen-
sion 3 because there are 3 elements
x
yz
J
JJ
. As quan-
tum mechanic acts on complex vector spaces, the mini-
mum dimension is a two-dimensional complex space or a
four-dimensional real space with two elements multiplied
by 1i

 which is different from a true four-dimen-
sional real space.
2.1. Matrix Description
The two components of the vectors of such a space can
be written as


123 4
,
x
ix xix
1
2
12 1
34 2
ie
ie
i
i
xx r
xx r









or in matrix notation:
(6)
C
opyright © 2013 SciRes. JMP

D. GRUCKER 931
which corresponds to an ordered set of four real ele-
ments.
An operator in such a vector space is given by
ij
M
a


12
34
viv
viv






 

2222
1234
vvv

a real 2  2 matrix multiplied by a complex
number α. It can be noticed that the two parts of α pro-
vide the information about the position of i in the four
real number set aij.
Quantum mechanics only studies operators which pre-
serve the norm, and in two dimensions this is written as:
1211 12
342122
vivaa
vivaa



 (7)
with
2 2 2 2
1234
v v v vv

   (8)
The solution is:


2

22
11 21 1aa
(9)

22
12 212 1aa
1221 220aa
 
†1MM
 

2
 (10)
11
aa (11)
which corresponds to

(12)
This result can be easily generalized to quantum vector
spaces of any dimension. We see that αM is not a unitary
matrix because it fulfills only one of the two conditions
of a unitary matrix


††
1UU
11 12
21 22
kk
kk
kk




UU . This implies that
quantum operators need not be unitary in order to pre-
serve the norm in the vector space contrary to frequently
held views.
The problem studied was to find the 12 real elements
of the three following matrices:
J (13)
with the three αk complex numbers for k = x, y, z. The
subscripts have presently nothing to do with the space
coordinates.
From Equation (1), it follows that the matrix Jz is:
11 12
21 22
xy zz
zz



12212112
yxy

12 2211
xyy

21 1122
xyy
zz
z
Ji

 (14)
with
11
zx
1212 1122
zyxx

zyxx
2121 2211
2221121221
zxyxy
which shows that 2211 . Mor generally it is well
known that the commutator of two matrices results in a
matrix with a null trace. As Jx and Jy are defined by a
commutator, we also have 22 11
x
x 11 yy and 22 .


From Equation (2) it follows that the matrix Jx is:
11 12
21 22
yz
x
x
x
J
x
x i






(15)
with
1221 111121121221
11 2
12 21
22
2
yyxy xyxy
xyy
 

2
12211211 11
12 22
1112 21
24
42
yx yyx
xyyy



2
21 12211111
21 22
1112 21
24
42
yx yyx
xyyy




12 21111112212112
22 2
12 21
22
2
yyxy xyxy
xyy


From Equation (3) it follows that the matrix Jy has the
same expression as Jx just by exchanging y and x. Calcu-
lating 21 121221
x
yxy

by eliminating x11 gives a relation
between the Jy elements:


2224
1112211221
42 40yyy yy

 (16)
As Jy is a quantum operator we have the general con-
straints given by Equations (9)-(11) on the Jy elements:
22
11 212
1
x
yy K


 (17)
22
12 112
1
x
yy K


 (18)


11 12210yy y


0y0yy
11 0y
(19)
where K is a positive real constant.
Equation (19) implies that or .
1112 21
For

, Equations (17) and (18) imply that
22
21 12,yyK 21 12
yy
11 0y
therefore .

and 21 12 yy, Equation (16) gives For

12 2yi and we obtain the second Pauli matrix


1
0
0
2
yy
i
JJ i





 (20)
or its opposite 2yy .
J
J


11 0yFor

and 21 12 yy

Equation (16) gives
12 2yi and we obtain the first Pauli matrix:


3
01
10 2
yx
JJ  



43yy
(21)
or its opposite
J
J


11 0y
The case

permits to calculate the constant
24.

D. GRUCKER 932
For and , Equation (16) gives
11 0y
222
21 12
yy
224
12
4 0y

11 12
42yy and using Equation (18)
222
11 12 we obtain
4yy2
12 0y

and 11 2y which
gives the third Pauli matrix:
5yz
JJ 10
01 2






65yy
(22)
or its opposite
J
J . The 6 Jyk matrices are Jy, −Jy ,
Jx, −Jx, Jz, −Jz.
The norm preservation and the null trace constraints
applied to Jx give the 6 different Jxk matrices Jx, −Jx, iJy,
−iJy, Jz, −Jz which fulfill the angular momentum criteria.
Let us note that all are real matrices.
All the matrix triplets which fulfill the angular mo-
mentum definition in the two-dimensional complex vec-
tor space are obtained with Equation (14). For each Jxk
there are 6 Jyk which gives 36 possibilities for Jz which
fulfill the 3 Equations (1)-(3). Out of these 36 possibili-
ties only 24 give a nonzero matrix. Obtaining the 24 so-
lutions is straightforward and only two examples will be
given explicitly. The first one is the well-known Pauli
spin 1/2 matrices:
11
01 0
,,
10 0 22
xy
i
JJ
i




1
1 0
0 1 2
z
J
 

 



 

(23)
and the second example is less conventional:
1313 13
01 01
,,
10 10 22
xy
JJ
 

 

 
 0
0 2
z
i
J
i






1 0
0 1
Z
J





(24)
It can easily be verified that these 3 matrices fulfill the
3 angular momentum definitions given by Equations (1)-
(3). The 24 solutions show that the elements of the spin
1/2 matrices can take only four values (−1, −i, i, +1)
times ħ/2 and the corresponding real matrices:
0101
,,
101 0
XY
JJ




 (25)
Let us note that:






22
1,
XY
JJ
2
1, 1
Z
J (26)
and






††
1, 1
Xx YY
JJ JJ
†
, 1
ZZ
JJ 
2222
(27)
which proves that all the solutions correspond to an op-
erator which preserves the norm of the vectors in the
two-dimensional space.
The 24 solutions of the general definition of the angu-
lar momenta for spin 1/2 are given in Table 1.
The general treatment of angular momenta uses the
operator
x
yz
J
JJJ. If we calculate this operator
for the 24 solutions we find 22
1
3
4
J
I in 16 cases
corresponding to the first two and last two rows of Table
Table 1. The 24 solutions for spin 1/2 using matrices JX, JY,
JZ, in ħ/2 units.
J1(JX,iJY,JZ) J2(JX,−iJY,−JZ)J3(JX,JZ,−iJY) J4(JX,−JZ,iJY)
J5(−JX,iJY,−JZ)J6(−JX,−iJY,JZ)J7(−JX,JZ,−iJY) J8(−JX,−JZ,iJY)
J9(JY,JX,iJZ) J10(JY,−JX,−iJZ)J11(JY,JZ,−iJX) J12(JY,−JZ,iJX)
J13(−JY,JX,−iJZ)J14(−JY,−JX,−iJZ) J15(−JY,JZ,iJX) J16(−JY,−JZ,−iJX)
J17(JZ,iJY,−JX)J18(JZ,−iJY,JX)J19(JZ,JX,iJY) J20(JZ,−JX,−iJY)
J21(−JZ,iJY,JX)J22(−JZ,−iJY,−JX) J23(−JZ,JX,−iJY) J24(−JZ,−JX,iJY)
1. This result is the well-known value obtained with
Pauli matrices. In the other 8 cases (rows 3 and 4 of Ta-
ble 1) we obtain 22
1
1
4
J
I
kkxxkyykzz
JJJ
.
We can associate the spin 1/2 with components Jkx, Jky
and Jkz to the space coordinates in the laboratory frame
(O,ux,uy,uz) to define the spin vector of a particle:


J
uuu
(28)
2.2. Dirac Equation
It is well known [2] that the spin of a particle can be re-
trieved using an equation which satisfies the special the-
ory of relativity and quantum mechanics postulates, such
as the Dirac equation. In Dirac’s book [3], the Pauli ma-
trices are obtained from the commutation definition with
the additional constraint:
2
222
4
xyz
JJJ

0e
e0
ia
xia





2
222
1, 0
,0
xxyyx
xx
mc


(29)
Owing to the fact that each observable has only two
eigenvalues ħ/2 and −ħ/2. He obtained
(30)
and stated that the phase
 could be adequately chosen so
as to obtain the three Pauli matrices. It is clear that Dirac
did not consider non observable spins and he only took
the 3 Pauli matrices as components of the spin vector.
In his relativistic theory of the electron, he found that
in order to obtain a linear wave equation it was necessary
to introduce four 4  4 matrices solving the following
equations:



1
1
0,
0
kk
k
kk
I
I
(31)
and the same relations obtained by permuting x,y and z.
A solution given by Dirac to these four matrices was:




 

 

 
(32)
with k = x, y, z, which nowadays is referenced as the
standard form. But it is well known that any set of ma-

D. GRUCKER 933
trices obtained with:
††
U UandUU



 



††
1UU
(33)
where U is any unitary matrix , is also
a solution to Equations (31). It can be shown that for a
homogeneous magnetic field, the energy of the interac-
tion of the intrinsic momentum with the magnetic field is
UU
0
2
s
q
EB
m

 (34)
and the spin definition can also be retrieved as
1
2
 S

(35)
where

is the spin vector defined by the 3 Pauli ma-
trices as in the non relativistic case. Dirac’s equation has
to be discussed more precisely only when time evolution
is introduced.
An important question is why does the general treat-
ment of quantum angular momentum give only one solu-
tion? This comes from the definition of 2222
x
yz
J
JJJ

1,,
which is a dot product and not the matrix product of two
angular momentum components. This definition intro-
duces some constraints which permit to eliminate the
solutions where J2 < 0, but by taking only the positive
values of j, also eliminates the negative components in
the combination of the Pauli matrices. Therefore only the
solution

x
yz
J


is obtained.
3. Angular Momenta for 3-Dimensional
Quantum System
If we consider the extension to three dimensions, a vector
has now 3 components and any quantum operator is de-
scribed by a real 3  3 matrix multiplied by a complex
number


1
2
3
00
00
00


. We can choose a basis for the vector space
where one angular momentum is diagonal. If we take the
z direction for this, the angular momentum is:
z
J






J
 (36)
where the λk are three complex numbers.
As the angular momenta are defined by Equations
(1)-(3) which correspond to non commuting matrices the
trace of the 3 angular momentum matrices Jx, Jy, Jz is null.
If we suppose that in the same basis and for the most
general case yij


 where βij are complex numbers,
we obtain according to Equation (2):





 
12 21
21 21
31 313232
0
10
x
Ji


 
 







13 31
23 32
0


 





(37)
and Equation (3) gives:
 


 
2
2 2
122113 31
2
2
21 2123 32
22
31 313232
1
0
0
0
y
J
 
 
 






 





(38)
and Equation (1):
4
1
z
ij
J







 
(39)





 

 
 

 

3 3
11122112133113
3
3
2212212123 3223
33
3313 3131233232
12133231 23 123
131223213 22 32
2321 131231231
21233132 1 3 1
2
2
2
2
2
2






 

 









  
  
23
31213212231132
3231121 32 1 231
2
2
2
(40)



 

 



0
To find the angular momenta, Equation (39) must be
equal to Equation (36). There are several solutions ob-
tained from the non diagonal Jz matrix elements equal to
zero.
If 1331 2

 , then 1. For 1




 we
obtain a set of three matrix solutions to the spin equation:
12
2
23
12
2
23
12
2
23
12
2
23
0i 0
10 0i
00 0,0i,
2
00 1
i
00
2
00
0
2
00
2
zx
y
JJ
J










































, (41)
For 1


 the sign of Jz and Jx is changed but Jy is
unchanged. If 12 23 we retrieve the con-
ventional matrices for a spin 1:
2i  

D. GRUCKER 934
10 0
00 0,2
00 1
00
0
200
zx
y
JJ
i
Jii
i

















010
101,
010





0
, (42)
Another solution is obtained with 23321

 
2
,
then

 and for 2

 the angular momenta are:
12 13
00 ,
00
ii 











2
12
2
13
12 13
2
12
2
13
0
00 0
01 0,2
001
2
1
00
2
00
2
zx
y
i
JJ
i
J




























, (43)
The third set of solutions is obtained with
12213 0

  1 , then


 and for 1


 the
angular momenta are:
13
23
22
23
0
0 ,
0
i
i












†
13
13
23
22
13 23
100 0
010, 0
000
22
00
00
0
22
zx
y
JJ
ii
J





 















, (44)
The choice of the basis of the 3-dimensional vector
space can also be made so as to have a diagonal angular
operator in the x or y directions, then the equivalent sets
of solutions will be obtained by permuting the x, y and z
subscripts. The important result of this derivation is that,
contrary to the 2-dimensional case, there is no solution
where the derived angular momentum operator preserves
the norm of the vectors in the 3-dimensional vector space.
Indeed, kk
J
J is never equal to 1.
Therefore, the spin of any boson in particle physics
requires a description other than the matrix representa-
tion of the angular momentum.
An interesting way to define the spin is to use the
framework of a Clifford algebra defined in the quantum
vectorial space.
4. Clifford Algebra and Spins
In his paper, in order to define the spin, Fröhlich [1] used
a Clifford algebra involving 2k  2
k matrices over com-
plex numbers, which has to be compared to the geomet-
ric algebra of spacetime initiated by Hestenes [4-6] and
also developed by the Cambridge astrophysical group
[7-9].
Grassmann and Clifford’s geometric algebra is based
on the definition of the geometric product uv for vectors
u, v, w obeying the following rules:


uv wu vw (45)


uvwuv uw (46)


vwuvuwu (47)
2
2
v
vv  
uv vu
(48)
where εv is the signature of v and the magnitude |v| is a
real positive scalar. The geometric product can be de-
composed into a symmetric inner product:


uv vu
(49)
and an antisymmetric outer product:

 
.uvu vuv
(50)
such that


111 1
222 2
10
10
uuu u
uuu u
(51)
Orthonormal vectors are defined by:




12
0uu
(52)
and


21u
(53)

The signature of orthonormal vectors uk are k
1212
1,, ,.
scalar, vector, vector, bivector.
uuuui

222
1212121 12
uuuuuuu u 22
12
1uu
.
The fact that real numbers can give negative squares is a
little surprising but it is well illustrated by matrix JY of
Equation (25).
4.1. Basis of Geometric Algebra in Two and
Three Dimensions
In two dimensions we have the geometric algebra of the
plane with four elements:
(54)
The bivector u1u2 has a property such that its square
. If the signatures



D. GRUCKER 935
or 12 . It can be shown that
12 forms a natural subalgebra equivalent to com-
plex numbers 12
. The bivector i is called a
pseudoscalar which anticommutes with vectors u1 and u2.

21uu 
zxyuu
122 3
22
121,uu
uu i
nxx n
An arbitrary linear sum over the four basis elements is
called a itmultivector:
01
A
aauauai 
a
i

221

22
22 2331
(55)
with components .
i
The sum of two multivectors is obtained by adding
each component and the geometric product is obtained
from the multiplication table (Table 2).
We can notice that the same algebraic properties are
obtained by using the {1, u1, u2, = u2u1} basis vectors
(where ~ indicates a reversion of vector products). Again
is equal to −1 if the signatures are
or .
 222
12
iuu
22
12
1uu 12
The product of two multivectors AB is explicitly given
by:
uu

00 111
A
Bababuabuab (56)


22
232 21
uabu u
01 10 23
ab abab


22
0311 2
babu u

03 1221 30
ab i


2 22 22 2
011 223
02203
1a au au a
auaai
  



222
012312
2 2
223
11, 0,
1
02 131 2
abu a
ab ab ab
ab
2
A
and the square of a multivector is
011
2aa u a (57)
Unit multivectors are characterized by:
22 2
011
or 0,
A
aaaa uu
aua
  
aau

(58)
Rotation in geometric algebra is based on a theorem by
Hamilton: given any unit vector n (n2 = 1), we can re-
solve an arbitrary vector x into parts parallel and perpen-
dicular to n:
x
xx
nxx n

u2 ui
. These components are identi-
fied algebraically through their commutation properties:
(59)
Table 2. Multiplication table for geometric algebra in two
dimensions.
1 1
1
u 2
11u i 2 12
uu
2
21
uu
1
2
u
i
i 2
21u
2
12
uu 2
21
uu

(60)




The vector
x
x


 can therefore be written −nxn.
Geometrically, the transformation
x
nxn represents
a reflection in the plane x, n. To make a rotation we need
two such reflections:
x
mnx nm (61)
with mn nm = 1 [7]. The multivector R ≡ mn is called a
rotor and

R = nm is called the “reverse” of R. R satis-
fying R


RR = 1 is defined as “unimodular”. R=
The bilinear transformation of vectors is a very general
way of handling rotations which can easily be general-
ized to vector spaces of any dimension [7].
In three dimensions we have the geometric algebra of
space with 8 elements:




123
123 123123
1, ,, ,,,,
1 scalar,1 trivector
3 vectors,3 bivectors,
uuu i
uuu uuuuuu

(62)
Again the pseudoscalar is defined by its
property such that 123123
iu
12 3
iuuu
21uuuuu

 , therefore the
signature of the orthonormal vectors must be one of the


222
123
1uuu

 or four possibilities:


22 2
12 3
1and1uu u

 or


22 2
13 2
1and1uu u

 or


22 2
231
1and1 .uu u


Contrary to the 2-dimensional case, varying the order
of bivectors does not give the same algebraic properties.
By choosing the bivectors:


1122 31323
,,uuuu uu  
222
123123
1 
(63)
for the basis of the 3-dimensional vector space, we have
the relation defining the quaternion algebra:

 
222
123
1uuu
(64)
which is only valid with the signatures:


0 11223341
52 63 7
1RABRRu RuRuR
RRRi
(65)
The important property given by Equation (64) is that
the trivector i commutes not only with vectors but also
with bivectors.
The multiplication table for 3-dimensional geometric
algebra with the signatures, in order to have the general
case, is given in Table 3.
The product of two multivectors is given explicitly by:



 (66)
with

D. GRUCKER 936
23
uu
Table 3. Multiplication table for geometric algebr a in three dimensions.
1 1
u 2 u 3 u 12 uu 31 uu i
1
u
u 21u1
uu
12
21u
uu 31
uu
uu 2 uu
12
2
uu
2 uu
13
2
123
uuu
21
2
23
uuu
2
312
uuu
21  22
123
uuu
113  22 132
uuu
3 22
231
uuu
 1
i
2
3
u
12
31
uu
2
23
uu
23
2
31u
i 2
uu
23
2
32
uu
1
i 2
31
uu
12
uu
uu 2 uu12
2
uu
2 uu i 22 1uu
12
2
uuu
2
123
22
1uu
uuu 2 uuu
23
1
2
uuu
3
2
uu
i 2
uu31
2
32
uu
12
3
2
23
1
uuu
13
2
312
uuu
312
22
23
1uu i 2
23
uu
i 2
uuu
12 3
2 uuu
23 1
2 uuu
312
22 uuu
12 3
22 uuu
132
22 uuu
231

222
000111222333
22
62377
22
53
22 22
3762 3
2
363
2
3 751
Rababuabuabu
buuab
u
u abuu
abu
u abu
 



22
3
0
2 22
2 7412
22
73
2
733
1
2
722
u
b
u abuu
u
abu
abu



2
2
711
abu
aa
22 22
441255136
101102423
2 2
42 253367 2
22
202141202
222
41157 1363
abuuabuua
Rabababuab
abu abu abu
Rababuabu
abuabuuab

 

 
 
22
30315126 23
22 2
4712 51162
404122123
22
40 563653
2
505132723
22
46 25064 2
2
6061712
Ra babuabua
abuuabu ab
Rabababuab
ab abuabu
Rabababuab
abuab abu
Rababuab
 

 
 
 


33
22
451 541 60
707162534
43 526170
ab
abu abuab
Rabababab
ababab ab




According to the previous mentioned Hamilton’s
theorem, any multivector A can be resolved in a com-
muting (perpendicular) part: Ca and anticommuting (par-
allel) part: Ja, therefore:
A
CJ
07
1
a
Ca ai
233415263
(67)
with
11 2a
J
aua uauaaa

  
because 1 and i commute with all the elements of a mul-
tivector.
If we look at the spin definition in natural units (ħ = 1)
given by the spin equations in a geometric algebra frame-
work, JxJy can be considered as anticommuting mul-
tivectors defining a rotation RZ in the x, y plane.
To prove the validity of this unconventional definition
of spin 1/2, we consider the first spin equation,
x
yyx z
J
JJJiJ which can be easily calculated with
Equation (66). It appears that the scalar and the pseudo-
scalar components of Jz are equal to zero. Since this rela-
tion applies to the other two spin equations, every Jx, Jy
and Jz have no scalar and no pseudoscalar components.
The first spin equation gives the following 6 equations:






22 2 222
353533422242 623
22 2222
14141163336351 3
222222
262622511 151412
222
1221 56365333
222
31 1364246222
23 32
2
2
2
2
2
2
x
yuxyuxyuxyuzuu
x
yu xyu xyuxyuzuu
x
yu xyu xyu xyuzuu
xyx yxyux yuzu
xyxyx yux yuzu
xy xy
 
 
 
 
 


222
451 54111
xyuxyuzu 
(68)
Calculating the components of the product of two mul-
tivectors
z
xy
RJJ

allows to retrieve the 6 previous rzk
factors and two new factors rz0 and rz7 for the scalar and
the pseudoscalar components of the product:
011122 33 41
52 637
zzz zzz
zzz
R rurururur
rrri




22222
0 111222333 4412
22 22
551 36623
7162534435261
z
z
rxyu xyu xyuxyuu
xyuu xyuu
r xyxyxyxyxyxy
 


(69)
with
therefore

07
07
1
12
1
12
z
xy zxyyxz
zz z
RJJrJJJJ ri
rriiJ
 


(70)
An analogous derivation gives:

07
07
1
12
1
12
z
yx zyxxyz
zz z
RJJrJJJJ ri
rriiJ
 



(71)
Therefore
222
07
1
14
z
zzzz zz
RRRRrriiJ
2
k
(72)
J
(with k = x, y, z) have only a scalar and a where

D. GRUCKER 937
pseudoscalar component. The important consequence is
that the 2
k
J
commute.
Hamilton’s theorem proves that there exists a unit vec-
tor which resolves any vector into a perpendicular and a
parrallel part and u3 is a good candidate to play this role
for Rz because u3 is perpendicular to any multivector Jx
and any multivector Jy. The permutation of x,y,z shows
that there is no incompatibility in doing so for JyJz and
JzJx. We retrieve the relation (65):
222
123
1uuu (73)
and find the 3 relations for

1
kk
RR 

:





22
22
22
1
11
4
1
11
4
1
11
4
z
y
x
i J
i J
i J



,,
07
07
07
zz zz
yy y y
xx xx
RR rr
RR rr
RR rr



(74)
Therefore a spin 1/2 (S1/2) can be defined in geomet-
ric algebra as a set of 3 rotors:
z
xy yz
RJJRJ
x xyz
JRJJ
5 26 3
3
1
(75)
As Rk are unimodular vectors, they preserve the mag-
nitude of multivectors. Equations (68) and (69) show that
the solution for the component values of the set of the 3
rotors depends on the signatures of the orthonormal vec-
tors.
In summary, in geometric algebra a spin 1/2 is a set of
3 rotations obtained from the product of two multivectors
which are the linear summation of 3 orthonormal vectors
u, u, u and 3 bivectors 
,

 ,

 according to:

11223 341
222
12212
,,
with
k
J
ku kukuk
kxyz
k k



 

123
 
 


(76)
which fulfill the spin equations:
x
yz
J
JuuuJ


 
0JJJ
0
xy yx
JJ
(77)
and the permutation of (x,y,z).
It can be noticed that the only solution if we try to
solve the spin equations in the two-dimensional case, is
xyz . This result is not surprising because in
two dimensions, rotations commute and
.
JJ
4.2. Spacetime Algebra
According to Hestenes [5], the standard model for space-
time is a real 4D Minkowski spacetime with vector addi-
tion and scalar multiplication where we can impose the
geometric product defined by equations 45 to 51 in order
to generate a geometric algebra called spacetime algebra
(STA).
A basis for STA can be generated by a frame


1234
of orthonormal vectors which determines
the pseudoscalar:
,,,uuuu
1234
iuuuu (78)

In order for the pseudoscalar i to keep the same prop-
erties as in the two- and three-dimensional cases, i2 must
be equal to −1, which imposes two signature combina-
tions which are called the metric of STA. One possibility
to fulfill i2 = −1 is that three signatures should be equal to
1 and the fourth equal to −1, the second possibility is that
three signatures should be equal to −1 and the fourth
equal to 1. The metrics are defined by

1,1,1, 1

and


1, 1,1,1
2 2222 1
xyzt
iuuuu
. The 3 orthonormal vectors with identical
signature will be associated with the space components
x,y,z and the other with the time component t so that:


11
22
33
xyyxxt tx
zx xzyt ty
yzzyzt tz
uuuuuu uu
uuuuuu uu
uuuuuuuu
(79)
By forming all distinct products of the four uk we ob-
tain a complete basis for STA consisting of 24 = 16 line-
arly independent elements.
There are 12 bivectors obtained from the arrangement
of 2 out of the 4 orthonormal vectors, but there are only 6
with different magnitudes:






 
 
 
(80)
As regards trivectors, there are 24 products to be ob-
tained from the arrangement of 3 out of the 4 orthonor-
mal vectors. There are 8 distinct products and only four-
having different magnitudes:
0
2
1
2
2
2
3
2
t
xyzxzy
t
x
ytz yzt
x
y
xt zx zt
y
z
xt yx yt
z
iu
uuuuuuu
iu
uuuuuu u
iu
uuu uuuu
iu
uuuuuu u


 
 
(81)




 



1234
1scalar :1
4vectors :,,,
6trivectors :3,3
4pseudovectors:
1 pseudoscalar:
uuuu
i


We can verify that i anticommutes with vectors and
trivectors but commutes with bivectors.
A complete basis for STA is:
(82)

D. GRUCKER 938
According to the 3-dimensional geometric algebra
case we have the quaternion relation (64) for the


bivectors. By adequately choosing the bivectors of the
vector space basis, the 3 new bivectors σk fulfill the fol-
lowing quaternion relation:
  
2 22
1231
iiii  
 
23
1ii  


2
3
3
1
t
t
u
u

  

1,1,1, 1
(83)
The four pseudovectors are: ξ0 the pseudoscalar of di-
mension 3 and the three other ξk which, again, fulfill a
quaternion relation between the geometric product of the
time orthonormal vector and ξk:


22
12
12
tt
tt
uu
uu


 (84)
The 3 quaternion relations are verified with the metric
and the choice of the basis defined by Equa-
tions (80) and (81).
For the other metric, Equation (84) is still valid, but to
keep the quaternion relations (63) and (83) for bivectors,
we have to consider another basis where
22

x
z, which corresponds to reverse rotations
in the x,z plane.
uu  

1 2 3
,,,
If we consider the geometric algebra definition of spin
1/2 given by Equations (76) to (77) we retrieve the spin
1/2 characterized by the 6-element multivector:
,,,1 ,,
x
yzx y
Suu
z z
u

 (85)
but in the 4-dimensional vector space there are two other
spins 1/2:



,,,21 2
2222
,, ,i ,i,i
,,, ,
xyzxyz
xyzzty
Suuu
uuu uuuu


3
22
123
,
t xt
uu


  (86)



,,,3,,,
,,,
xyzx y z
xyzt
Suuu
uuuu

1 23
222
123
, ,
, ,
t tt
t t
uuu
u u





1,1,1, 1

(87)
for the metric . For the other metric with the
other basis there are three new spins 1/2 obtained from
the 3 vectors ux, uy and uz:


1 2 3
,,,
z
u
,,,4 ,, xyzx y
Suu



(88)


22 22
,,,5 1
,,, ,
xyzxyzzty t
Su u uuuuu22
2 3
,xt uu

 

2
2 3
, ,
t t
uu
(89)

22
,,,6 1
,,,
xyzx yz t
Suuuu

 
,,,1..6,, ,1..6
,,
ztyzt
SSS

444 444444
1122 33 4451
62738192 103 110
12 1132143
,,
with
,,
(90)
There are 18 other spins 1/2 obtained with the different
vectors of the 4-dimensional case and the same bivectors:
,,,1..6xyt x (91)
Therefore the 24 spins 1/2 found with the matrix deri-
vation are retrieved in the STA description but here with
a clear distinction between spins 1/2 which fall in two
metric-dependent categories. If the 12 spins 1/2 of each
category were sorted as a function of bivectors we would
find 3 families of 4 spins 1/2 as in elementary particle
physics.
As the definition of rotation given by Equation (61)
applies whatever the dimension of the vector space, in
four dimensions the spin 1 can be defined by set of 3
rotors:
z
xyyzyx yz
k
RJJ RJJ RJJ
Jkukukukuk
kkkkk k
kkk kxyz




  
 
 
2221uuu
(92)
where the Jk fulfill the spin equations. From the gener-
alization to 4 dimensions of the vector product given by
Equation (66), it is easy to see that the scalar component
is null because real components commute and the pseudo
scalar component is null because its parts are independ-
ent of the signatures. As the Rk are unimodular vectors,
they induce rotations which preserve the norm of any
vector. Again the solution to the equations which deter-
mine the components of the multivectors Jk is dependent
on the signature of the orthonormal vectors. Therefore
there are 4 different solutions with space-like signatures
xyz
 21u and time-like signature t


 and
12 different solutions if we consider each signature in-
dependently.
5. Conclusion
We have shown that for the two-dimensional complex
vector space, the spin matrices can be calculated directly
from the angular momentum commutator definition. We
have retrieved the 3 Pauli matrices and found 23 other
triplet solutions. When extended to the three-dimensional
space, we have shown that there is no matrix which pre-
serves the norm of the vectors and fulfills the spin equa-
tions. By using a geometric algebra with a vector prod-
uct which combines a commuting product and an anti-
commuting product it has been possible in four-dimen-
sional spacetime to retrieve the 24 different spins 1/2
defined as 12 clockwise and 12 counter-clockwise rota-
tions. These rotations are characterized by anticommut-
ing parts composed of 3 vectors and 3 bivectors which
fulfill the spin equations. Spin 1 can be defined as 3 rota-
tions characterized by 4 vectors, 6 bivectors and 4
trivectors which fulfill the spin equations. These uni-
modular spin 1 rotations preserve the magnitude of mul-
tivectors. There are 12 different spins 1 depending on the
signature of the 4 orthonormal vectors of the four-di-
mensional vector space. The correspondence between
this derivation and particle physics is perhaps fortuitous
but the use of STA offers the advantage of formulating

D. GRUCKER 939
conventional relativistic physics in invariant form with-
out reference to a coordinate system [5] and it seems
promising to analyze how time evolution and spin inter-
action can be used in order to predict the gyromagnetic
ratio of the proton and the neutron.
6. Acknowledgements
I would like to thank Jacques Baudon, Edward Belaga,
Jules Grucker, Tarek Khalil, Jean Richert, Kees van
Schenk Brill for enlightening, and Nathalie Heider for
liberating my mind.
REFERENCES
[1] J. Fröhlich, “Spin or Actually: Spin and Quantum Statis-
tics p1-60 in The Spin, Poincaré Seminar 2007,” B. Du-
plantier, J. M. Raimond and V. Rivasseau, Eds., Birk-
häuser, Basel, 2009.
[2] C. Cohen-Tannoudji, B. Diu and F. Laloë, “Quantum Me-
chanics,” Wiley, New York, 1977.
[3] P. A. M. Dirac, “The Principles of Quantum Mechanics,”
Oxford University Press, Oxford, 1930, pp. 157-162,281-
285.
[4] D. Hestenes, American Journal of Physics, Vol. 47, 1979,
p. 399. doi:10.1119/1.11806
[5] D. Hestenes, American Journal of Physics, Vol. 71, 2003,
pp. 691-714. doi:10.1119/1.1571836
[6] D. Hestenes, American Journal of Physics, Vol. 71, 2003,
pp. 104-121. doi:10.1119/1.1522700
[7] S. Gull, A. Lasenby and C. Doran, Foundations of Phys-
ics, Vol. 23, 1993, pp. 1175-1201.
doi:10.1007/BF01883676
[8] C. Doran, A. Lasenby and S. Gull, Foundations of Phys-
ics, Vol. 23, 1993, pp. 1239-1264.
doi:10.1007/BF01883678
[9] J. Lasenby, A. N. Lasenby and C. J. L. Doran, Philoso-
phical Transactions of the Royal Society A, Vol. 358, 2000,
pp. 21-39. doi:10.1098/rsta.2000.0517